Ionizing Radiation Interaction with Matter

[Graham87]

Homework Statement

Problem c)

Relevant Equations

Beer Lambert Law
$\phi(x) = \phi(0) e^{-\mu x}$
$T=e^{-\mu x}$

Basically, I don't understand how they got x=L/2 and x=r paths of T_in and T_out. Below is elaboration.

I don't understand the choice of paths for incoming and outgoing photons in the solution for c).
The solution put incoming path to x=L―/2, while outgoing x=r.
As I interpret it, T_in is the transmission factor before scattering, while T_out is the transmission factor after scattering?
The average photon path traveled in the cylinder is L― and we should divide by 2 for the average path L―/2 before scattering?
Why put the outgoing photons as r? So we assume the photons scatter in the center? But then again we assumed the incoming path is L―/2, which is an average that does not considers all traveling to the center.

So the incoming photons are not assumed to have a mutual path towards the center, while all the outgoing photons should be assumed to travel from the center?

Basically, I don't understand how they got x=L/2 and x=r paths of T_in and T_out.